The present invention relates to an integrating module for sampling control devices that can be fed back and in which the relationship between the input and the output signal at any sampling point is determined by a recursive integrating algorithm which is formed by approximation of the area under a continuous function by means of the trapezoidal rule.
The use of process computers and microprocessors in the control and automation of technical systems which are characterized by a high processing rate and a large storage and address space forms the technical foundation for the realization of high performance control systems. A prerequisite for the full utilization of such computers, however, is a sufficiently accurate physical/mathematical description of the loop to be controlled, for instance, in the form of a so-called "model". As is well known, the dynamic behavior between the inputs and outputs of a loop can be described in general by a set of differential equations of different order and of ordinary equations which were derived by utilizing the physical laws occurring in the respective system.
For the uniform mathematical description of such sets of equations and their formal standardized treatment up to the design and layout of suitable control strategies based thereon essentially two methods have evolved in modern control engineering. One of them is the well-known "frequency domain method", in which the system equations are transferred from the time domain into the frequency domain. The dynamic system behavior can be further processed there in the form of complex transfer functions up to the design and layout of suitable control systems.
A second method for the uniform mathematical description which has not been popular generally yet is the system representation in the so-called "state space." There, the system equations which as a rule are of different order are transferred in the time domain into a first-order set of differential equations by a suitable definition of the state variables. All input, output and state variables of the system can them be combined advantageously in vectors which are generally interlinked via a so-called "system, input, output and pass-through matrix." Besides the simple mathematical processing of such a system description by means of matrix calculus, it is a further advantage of the state space representation that the state equations, assuming linearity and time invariance, can be converted into so-called "normal forms" by means of standardized transformations. While these describe the same physical system and thus represent equivalent descriptions of the respective dynamic system behavior, certain structure properties of the system to be modulated, for instance, their eigen values, their controllability or observability, emerge particularly clearly only after the transformation of the state equations into one of the standard forms. The standard forms thus are a particularly suitable starting point for design of control strategies or control structures. Thus, for instance, the actual state of a concrete system can be simulated in an "observer" which "runs along", for instance, in a computer, and is modelling the system so that this "internal" state of the system becomes accessible and can be influenced by means of suitable control interventions: known standard/forms are, for instance, the so-called "Jordan standard form" and various "observer and control standard forms."
The description of the system by means of state equations or its representation in a selected standard form has the further advantage that it can be realized particularly well in analog and digital computers. This system description has therefore a decisive importance for a practical realization of technical sampling control devices. The good handling capability is made possible particularly by the provision that the normal forms lead to a standardized, block-oriented modular system representation. The number of such "blocks" required for the modelling of a technical system is directly dependent on the order of the system present, so that matching of the scope of the standard form required in each case to a changed "system order" is possible in a simple manner by adding or omitting one or several blocks. A further advantage in the technical realization of this system description is that all blocks are designed identically in one of the standard forms and are given in the core one integrator each in the representation in the time domain. Thus, for instance, a system model designed in accordance with the Jordan standard form consists of the parallel connection of a number corresponding to the respective system order of integrators provided with coefficient setters which can be fed back separately. The complex system behavior is therefore simulated by the superposition of the reaction of first-order subsystems. Accordingly, a system model designed according to an observer or control standard form consists of the series connection of a number of integrators corresponding to the respective system order and which are all fed back together via coefficient setters. Here, too, the complex system behavior is composed of the reaction of first-order subsystems.
It is known to simulate in the practical technical design of such system models or of "observers" based thereon the integrator required as a component in a digital control and automating device by means of a recursively operating algorithm. In the process, the actual values of the input variables are determined at each sampling point and therefrom, actual output values are determined with the aid of variables which were determined, calculated and interim-stored, at the preceding sampling point. In a known algorithm for simulating an integrator, the area under a continuous time function is approximated by means of the so-called "rectangle rule." For the latter, the relationship applies v.sub.k =v.sub.k-1 +T.sub.A /T.sub.I .multidot.U.sub.k-1 with
v.sub.k : output signal at the k.sup.th sampling instant PA1 v.sub.k-1 : output signal at the (k-1).sup.th sampling instant PA1 u.sub.k-1 : input signal at the (k-1).sup.th sampling instant PA1 T.sub.A : sampling time PA1 T.sub.I : integrating time constant PA1 T.sub.A : sampling time PA1 T.sub.I : integrating time constant PA1 v.sub.k : output signal at the k.sup.th sampling instant PA1 v.sub.k-1 : output signal at the (k-1).sup.th sampling PA1 instant PA1 u.sub.k : input signal at the k.sup.th sampling instant PA1 u.sub.k-1 : input signal at the (k-1).sup.th sampling PA1 instant PA1 an internal rectangle rule integrating module, in which the relationship between the input and the output signal at every sampling instant is determined by a recursive integrating algorithm which is formed by the approximation of the area under a continuous function by means of the rectangle rule, PA1 a first amplifier which weights the input signal of the integrating module with a first factor d.sub.o =T.sub.A /2.times. T.sub.I, where T.sub.A =sampling time and T.sub.I =integration time constant and makes it available as the input signal for the internal rectangle rule integrating module and as a pass-through value; a first adder which forms as the output signal of the integrating module the sum of the pass-through value and twice the output signal of the internal rectangle rule integrating module; PA1 a separate feedback output at which twice the output signal of the internal rectangle rule integrating module is delivered; and PA1 means for correcting the deviation of the signal at the feedback output from the output signal if feedback from the feedback output to the input of the integrating module is present.
which can be found, for instance, in the book by Norbert Hoffmann, "Digitale Regalung mit Mikroprozessor", Vieweg Verlag, 1983, on page 23, bottom. A "rectangle rule integrating module" operating recursively according to the equation above has the advantage that it can be fed back and thus can be used as a component, for instance, in the above-described standard forms in the representation of state space of technical systems. The rectangle rule integrating module, however, has on the one hand the disadvantage that it carries out the I- approximation only with a permanent mean area error. In particular, it is a further disadvantage that, for instance, when such an integrating module is set in operation at the sampling point k-1 (v.sub.k31 1 =u.sub.k-1 =0), an output signal v.sub.k is available only after the end of a sampling time T.sub.A at the following sampling instant k by weighting the input value u.sub.k-1 which was sampled for the first time at the preceding sampling instant k-1 with T.sub.A /T.sub.I. This in turn has the disadvantage, for instance, in an observer constructed according to a standard observation or control form, that a change of the input signal has traversed all series-connected integrating modules only after a number of sampling times corresponding to the order of the system and has "arrived" completely at the output of the observer. In a system model designed in this manner, a considerable reaction time must therefore be expected. It has been found that, for preserving the stability, the minimum system time constant to be simulated of the rectangle rule/integrating module must be larger at least by a factor 2 than the sampling time T.sub.A present in every case.
For dynamicizing such a modular automating system, two approaches are available. For one, the sampling time can be reduced so far that sufficiently small integration time constants can be adjusted while preserving the above-mentioned ratio to the actual magnitude of the sampling time T.sub.A. It speaks against such a procedure that the sampling time of a sampling time control system is limited by its internal design and organization, and such a system is already operated as a rule with the longest possible adjustable sampling time for reasons of accuracy.
Another approach for dynamicizing a sampling control system consists of the use of more powerful algorithms which permit a more advantageous ratio of the system time constant or integration time constant to the sampling time present. Particularly with a system description by means of the state space methods, this means the use of an improved algorithm for the numerical integration. Thus, it is known to approximate the area under a continuous function by means of the so-called trapezoidal rule. An integrating algorithm formed therefrom is considerably more powerful than the known rectangle rule algorithm and is described, for instance, in the book by Wolfgang Latzen, "Regelung mit dem Prozessrezhner", P.I. Wissenschaftsverlag, 1977, particularly on pages 79 to 91.
Thus, the relationship EQU V.sub.k =V.sub.k-1 +d.sub.o .multidot.(u.sub.k-1 +U.sub.k)
applies for the integrating algorithm according to the trapezoidal rule where d.sub.o =T.sub.A /2.multidot.T.sub.I
It can be seen from this equation that also with a start of the algorithm, for instance, at a sampling point k, an output value d.sub.o .multidot.u.sub.k can be expected immediately although at this instant no values v.sub.k-1 and u.sub.k-1 from a preceding sampling are yet available. In contrast to the integrating algorithm according to the rectangle rule, an integrating algorithm according to the trapezoidal rule "reacts" considerably faster to changes of the input signals. It can further be shown that the mean area error in numerical integration by means of the trapezoidal rule is always equal to zero while in numerical integration by means of the rectangle rule, it increases with the sampling time.
A condition for the use of integrating modules in a modular system model which is constructed, for instance, in accordance with one of the known standard forms is the capability of random wiring and in particular, feedback. An integrating module constructed according to the trapezoidal rule which operates recursively on the other hand cannot be fed back. The reason for this is that a change of the input value has a direct effect as a so-called "pass through value" without delay and thus, via the feedback, again without delay on the input value and so forth. If it is attempted nevertheless to feed back a trapezoidal rule integrating module, an equation is obtained which itself requires, for the calculation of the actual output value at the respective sampling point, this output value still to be calculated. Thus, no logical and feasible algorithmic sequence in the processing of the individual terms of this equation is obtained. If such an element is forcibly fed back in a computer, the recursive calculation would abort immediately. A basic condition for discretizing a continuous system by an algorithm, however, consists exactly in the fact that it can be calculated recursively at the sampling points.